#189276

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#380
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

815 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,9,6,4,13,8)(3,7,12,15,11,10,16,14)(17,18,22,27,23,28,21,26)(19,29,31,38,33,41,32,24)(20,25,34,39,36,40,35,42)(30,43,45,48,46,47,44,37);(1,3)(2,7)(4,10)(5,12)(6,11)(8,14)(9,15)(13,16)(17,19,21,32,23,33,22,31)(18,24,27,29,28,38,26,41)(20,30,35,44,36,46,34,45)(25,37,39,43,40,48,42,47);(1,4)(2,6)(3,10)(5,8)(7,11)(9,13)(12,14)(15,16)(17,20)(18,25)(19,30)(21,35)(22,34)(23,36)(24,37)(26,42)(27,39)(28,40)(29,43)(31,45)(32,44)(33,46)(38,48)(41,47);(3,11)(7,10)(12,16)(14,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,36,35)(24,29,38,41)(25,39,40,42)(30,44,46,45)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,d^c=dh,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96472);